Salamanders
is a thirty-piece wooden sculpture that was group assembled by thirty volunteers
in an exciting sculpture "barn raising" when I was artist-in-residence
at M.I.T. in October/November 2003. It is composed of laser-cut salamander-shaped
components which lie in the planes of a rhombic triacontahedron and were
mathematically designed to weave through each other and exactly fit together
on the outside.

1. Introduction

M.C. Escher playfully incorporated chameleons and other
reptiles or amphibians in his two-dimensional geometric artwork [1].
In homage to his creative spirit, I designed my sculpture Salamanders
to feature flat salamanders which interweave in three dimensions.
Figure
1 shows it hanging temporarily inside a window overlooking the construction
of Frank O. Gehryís new Stata Center at M.I.T. [2],
where the sculpture will eventually reside.

Figure 1:Salamanders

My ultimate concept, if funding can be
found, is for a large metal double sphere as shown in Figure
2. The inner and outer spheres are each made of thirty identical two-headed
salamander shapes. Each part is parallel to an identical part similarly
oriented in the other sphere. I find it visually interesting to show that
the same salamander parts can be joined in these two contrasting arrangementsóone
very open and one very interlocked. It is a puzzle with two very different
solutions.

The outer sphere of Figure
2 does not present any inordinate construction challenges. I am certain
that I can fabricate its thirty components and assemble them. The inner
sphere was my fundamental concern. From a computer rendering such as Figure
3, I can verify that there exists a configuration in which the parts
do not intersect each other in the interior, yet exactly meet edge-to-edge
along the exterior. However there is no mathematical method to determine
if the thirty initially separate rigid components of this sculpture can
be physically manipulated to weave them into the desired configuration.
What is the assembly algorithm? Notice that one could not simply position
pieces one at a time, because if one tried to insert the last piece after
all the others are positioned, the legs would block access.

The two hands of one sculptor are not
sufficient to manipulate so many components simultaneously, so this was
an ideal question for the collective creativity of a group assembly project.
I have led other sculpture "barn raisings" [4], but for
them I had a proven assembly algorithm pre-designed. When I was invited
to be artist in residence at MIT [6], this thirty-component
assembly project struck me as an ideal group activity to try there.

Although geometric and salamander-filled,
in most ways this work involves very different design problems than Escherís
art, because these parts interweave in three dimensions. Both spheres of
Figure 2 fall within a large family of geometric sculptures
I have been exploring [5], based on symmetrically arranged
planar components. Physically, these works consist of interwoven identical
components that can be accurately laser cut, delivered flat to the assembly
site, then woven through each other on-site and fastened together. Mathematically,
the design of these sculptures involves drawing within the "stellation
diagrams" of polyhedra. Other software exists for creating stellated polyhedra
[9], but my approach [5] is unique
in allowing design and visualization of free-form drawings within the stellation
planes. Stylistically, this allows me to create an Escher-like sense of
structured confusion.

For Salamanders,
the underlying polyhedron is the rhombic triacontahedron (RT), which consists
of thirty "golden rhombi," arranged with icosahedral symmetry, as shown
in Figure 4. When the thirty face planes of the RT are
extended to infinity, the pattern of their intersection in any one of the
planes is a set of lines partly visible in Figure 5.
Superimposed over these lines is the shape I drew for the salamander part.
The dark lines are to be laser cut and the light inner lines indicate how
the area is divided into triangles.

Figure 4: Rhombic triacontahedron (30 rhombi)

Figure 5: Salamander layout in RT stellation diagram

If made in metal plate, I would design
tabs with holes attached to the feet, folded to the proper dihedral angle.
Dotted lines in Figure 6 show how two foot tabs could
connect on the back of the head. Round-head bolts with hexagonal sockets
would be used to hold everything together and also serve as eyes. However,
for joining the wood components, I pre-made sixty small wood connectors,
miter-cut at the appropriate face angles and dihedral angles. The feet
screw into a connector glued behind each head.

Figure 6: Design with bolts for eyes

Figure 7:Solid freeform fabrication model

A 3D model of the form is shown in Figure
7. It is a 2.5 inch diameter model made on a Stratasys/Objet Eden 333
machine. The .stl file that describes its geometry is available at my web
site for anyone with solid freeform fabrication equipment who wishes to
make a copy [3]. Such models are assembled in thin cross-sectional
layers on a 3D printing machine, so they give a visual and tactile sense
of the structure. But they do not demonstrate whether the form could be
assembled from rigid planar components.

I was afraid to put myself in the embarrassing
position of creating thirty large parts and bringing a crew of thirty people
together only to find out my design was impossible to assemble. So I made
a seven-inch prototype in rigid acrylic plastic (Plexiglas), to verify
that the parts can be woven without jamming together. Before I had thirty
four parts cut, I had to lay them out for the 24 inch square working area
of a laser cutting service. Interestingly, the parts pack together quite
tightly, minimizing material wastage, as seen in Figure 8.

It took half a day to assemble the plastic
parts. By wiggling parts slightly to open small spaces which let other
parts be inserted, all the while holding everything loosely together with
many fingers, bits of tape, and small rubber bands, eventually I managed
to get the last piece in place. So I had an existence proof that assembly
was possible, but I certainly did not have anything like a well-defined
assembly algorithm. The resulting model, after gluing the parts, is shown
in Figure 9. Because it is clear, it may difficult to
determine what is what in that photo, but I can attest that in person it
is quite cool looking.

Figure 8: Layout for laser cutting plastic parts

Figure 9: Plexiglas model, 7 inches

3. Sculpture Barn Raising

When the time came for my residency at
MIT, I brought the acrylic model and the 3D printing model with me to show
people what we would be building at the group assembly event. I am very
grateful to my host Erik Demaine, and also to Marty Demaine and Abhi Shelat,
all of whom spent many hours with me on the preparations. In the days there
before the assembly, we used a laser cutter to cut the thirty wood salamanders
(and some spares). The wood we finally selected is a Baltic birch plywood,
laser-engraved with ovals for the eyes, then sanded, drilled and countersunk
in four places for screws, glued to two mitered wood junction pieces, and
given a protective coating of tung oil. We also made a large quantity of
scaled-down paper salamanders for practice assembly. One important lesson
we learned late along the way is not to purchase large amounts of expensive
ash plywood of a type which chars into embers when one tries to laser-cut
it.

If you want to make your own paper or wood model, Figure
10 is an image of how we laid out a pair of parts in the 32-by-18 inch
bed of the laser cutter we used. Even without a laser-cutter, you can make
thirty copies (enlarged) on card stock, cut them out with scissors, and
assemble your own paper model.

Figure 10:Layout for cutting wooden
parts

Figure 11: Taped together paper model

On the day of assembly, roughly fifty
volunteers started work around four large tables in one of the studio art
rooms. Everyone received an envelope with thirty laser-cut paper salamanders.
After some instruction on how the parts go together, and by studying the
models I brought, we began individually or in small groups to assemble.
Figure 11 shows one of the paper models. Figure
12 shows its construction. Small pieces of clear tape hold a long leg,
a short leg, and a head at each of the sixty junctions. It is important
to understand that the two-headed salamanders can have their heads all
facing to the left or all facing to the right, and that each salamander
is part of two different types of pentagonal cycles.

Figure 12: Working on the paper models

It is initially tricky to master how the
parts weave through each other, and internalize what should be inside and
what should be outside. The long legs which make a star pattern at the
five-fold junctions are the biggest problem. The "ankle" of each long leg
must be outside of the "knee" of the leg it crosses. Paper is flexible,
allowing legs to be bent around each other and into place. But the real
challenge for everyone was to design an assembly strategy which would later
work with rigid wooden components. After an hour and fifteen minutes of
paper practice, I felt that enough people understood the structure. We
then started on the real assembly of the wooden parts.

Figure 13: Starting the assembly of the wooden salamanders

Figure 14: Looking up into the first cap of ten salamanders.

After some discussion, the assembly method
we chose was based on three units of ten salamanders each: two polar caps
and an equator. Figure 13 shows the initial step, working
in the air with five parts around an imagined vertical five-fold symmetry
axis. Then another five parts weave into those, making the first cap of
ten parts. Figure 14 is a view from below looking up
into the first cap being assembled. It was loosely screwed together, and
then put aside. A second cap was assembled in the same manner. Our strategy
was to make an equator of the remaining ten parts to connect the top and
bottom caps.

Figure 15: Adding the equator to the bottom ring.

Figure 16:Finishing the equator

We did the final assembly by placing one
cap facing up on the table and first adding an equatorial ring of five
salamanders whose bodies are aligned vertically, as seen in Figure
15. Then we wove in the overlapping equatorial ring of five salamanders
whose bodies are aligned more horizontally, shown in Figure
16. (Each salamander is part of one ring of each type; each five-fold
equator is made of both types of ring, concentrically arranged but not
contacting each other.) Finally, the other cap was lowered on to the top,
as shown in Figure 17. This last step took some time
and a number of retries to get the legs properly interwoven.

Figure 17:Adding the top cap

Everything worked out, so the remaining
screws could be inserted and all 120 screws were tightened. The final result
is in Figure 18. I found it to be surprisingly rigid
for its 15 pounds of weight. The total time for the assembly of the wooden
parts was one hour and forty minutes.

Figure 18:Finished!

4. Conclusion

I consider the Salamanders
barn raising to have been a great success. I am very happy with the final
sculpture and the wonderful preparation and assembly support that I received
at MIT. Many more photos plus short digitized videos of the assembly are
available online [7]. The work has been selected as the
first in the art collection of the MIT Computer Science and Artificial
Intelligence Laboratory. Our plan for its future is that it will hang in
the new Stata center. After it is moved to its public location, I hope
Salamanders will remind viewers of
the beauty of geometry and the tradition of M.C. Escher.

I thank the many people at MIT who made
this project possible. First and foremost, Erik Demaine invited me to be
artist-in-residence through the Office for the Arts and the EECS department.
Erik Demaine, Marty Demaine, and Abhi Shelat spent many hours with me doing
the preparations, especially laser-cutting the wood and paper. Rebecca
Frankel helped with the oil finish. The CSAIL fabrication shop and staff
provided the laser-cutter and other resources. Erik Demaine, Mark Hoffman,
and Moses Liskov took many photos, including the ones in this paper. Tom
Buehler from the CSAIL Computer Graphics group took and edited video. Michele
Oshima, Nicole Ackerman, and Marc Rios at the MIT office for the Arts did
advertisement and behind-the-scenes logistic arrangements. Tech Talk ran
nice articles which helped draw a crowd to the event and report the results
[8]. And of course, my thanks most of all go to each of the barn raisers
who came and participated in the assembly.